Back in 2016 me and some collaborators used perturbation theory to calculate the stable modes in a nonlinear PDE that describes lasers. see "appendix C" on p17

Back in 2016 me and some collaborators used perturbation theory to calculate the stable modes in a nonlinear PDE that describes lasers. see "appendix C" on p17 of the pdf in https://arxiv.org/abs/1611.01676 https://arxiv.org/abs/1611.01676 The basic idea is that we started with a basis of solutions to the linear equation (d=0), then expanded the nonlinear equation order-by-order in the small parameter d. The math was pretty messy since it was a system of linear equations (involving both E and B fields) but conceptually it was pretty straightforward. It was pretty cool, and definitely very satisfying when the resulting eigenvalues agreed nearly exactly with numerical simulation.